Harmonic Moments for Supercritical Multi-type Galton–Watson Processes

In this paper, the convergence rates of harmonic moment E[(1 · Z_n)^−r] of a supercritical multi-type Galton–Watson process {Z_n; n ≥ 0} are studied. It is shown that there exists a phase transition in the convergence rate of the harmonic moments, which extends the results from single type cases to multi-type cases. Based on above result, under some moment hypothesis on Z_n, the large deviations of \(\boldsymbol{l} \cdot \boldsymbol{Z}_{n+1}\over \boldsymbol{1}\cdot\boldsymbol{Z}_{n}\) $$\frac{\mathit{l}\cdot {\mathit{Z}}_{n+1}}{\mathbf{1}\cdot {\mathit{Z}}_n}$$ are studied for l > 0. In particular, in the case that the offspring distributions are of Pareto-type, an explicit large deviation result is obtained.